Are there any particular term for a pair $(U;A)$ where $U$ is a set and $A\in\mathscr{P}U$? That is, saying informally, $(U;A)$ is a set $A$ together with a set $U$ on which $A$ is defined ($A$ is defined as a subset of $U$).

But normally, one does not use this terminology for the simple situation of "a set and a subset". You can view it as an element of the inclusion relation, but there are few contexts where this would be helpful.